I think that level creators should test and complete all their levels before publishing. When I start playing a level, I should be able to assume that there is a solution (no matter how difficult), but that the trick is to find it.
On the whole I agree with you. Good play testing is an essential step prior to the release of levels for any game.
When I was setting up the alphabet set building the letters was relativley quick. Most of the time was spent solving each of them in a couple of different ways to get an idea of the possible solution space, tweaking the difficulty and then setting the budgets.
That being said Unpossible is a special case. Its name and the fact that the budget is effectivly unlimited made it clear (at least to me) that the author wasn't sure if it could be solved at all.
I like the chalenge of trying to achieve somthing that you aren't 100% sure is possible, and the sense of triumph for whoever eventually solves it first will be that much sweeter.